# Concentration Inequalities

Bounds showing that a random quantity stays close to its expectation with high probability.

**Rests on:** Sampling Distributions

*Mathematical Foundations · Probability Theory*


## Why learn this

This concept is the mathematical backbone of why machine learning works at all. When you train a model on data, you are estimating something from a sample — and concentration inequalities tell you how much you can trust that estimate. They are the reason we can say 'with enough samples, the model's error on new data will be close to its error on the training data'. Without them, there would be no theoretical guarantee that a model generalizes beyond the data it saw. This unlocks: statistical learning theory, which is the branch of mathematics that analyzes why models generalize; and it deepens your understanding of any AI system that makes decisions under uncertainty — which is every AI system. For example, when an LLM is trained on billions of tokens, concentration inequalities are what justify the claim that its performance on the training data is a good proxy for its performance on unseen text. When you build a recommendation system, they tell you how many user ratings you need before you can trust the average rating. This is the foundation for the 'generalisation bounds' you will hear about in any serious discussion of model performance.


## The intuition

Concentration inequalities answer a simple question: when you average a bunch of random numbers, how close is that average to the true average you would get with infinitely many numbers? Suppose you flip a coin 100 times. You expect about 50 heads, but you might get 45 or 55. Concentration inequalities say: the probability that you are far from 50 (say, more than 10 away) is small, and they give you a specific number for that probability. The 'inequality' part means they give you an upper bound on the probability of being far from the expectation. They are called 'concentration' because they show that the average 'concentrates' around the true value as the sample size grows. The key insight is that the bound depends on two things: how many samples you have (more samples → tighter bound) and how spread out the random numbers are (if they are all between 1 and 2, the average is very predictable; if they range from 0 to 1000, the average is less predictable). These inequalities do not require you to know the exact distribution of the random numbers — they work for almost any distribution, which is why they are so powerful in machine learning, where you almost never know the true distribution.


## An analogy

Imagine you are trying to estimate the average height of all adults in India. You cannot measure everyone, so you take a random sample of 100 people and compute the average. You know that your sample average is probably close to the true average, but how close? The concentration inequality is like a guarantee from a mathematician: 'I cannot tell you exactly how close, but I can tell you that the probability of being off by more than X cm is less than Y.' The more people you measure (larger sample size), the smaller Y becomes. Also, if you know that heights in India have a small spread (most people are between 150 and 180 cm), then Y is smaller than if heights were wildly variable (say, from 50 to 250 cm). The analogy works well, but it has a limit: the guarantee is probabilistic, not absolute. It does not say 'you will definitely be within X cm', it says 'the probability of being off by more than X cm is at most Y'. For example, if Y = 5%, that means there is a 5% chance the estimate is bad. That 5% is not an excuse — it means you need to be aware that sometimes your sample might be unlucky, but the inequality tells you how often that happens.


## Definition

A concentration inequality is a mathematical bound on the probability that a random quantity (such as the sample average) deviates from its expected value by more than a certain amount, stated as a function of the sample size and the variability of the underlying random variables.


## How it connects

This concept sits at the heart of probability theory, which you are learning about. It builds directly on the idea of expectation — the average you would get with infinite samples — and variance, which measures spread. You have notes on Sampling Distributions, which describe how a statistic like the sample mean varies across repeated samples; concentration inequalities give you quantitative bounds on how far that sample mean can stray from the true mean. This also connects to the Empirical CDF, which is a way to estimate a distribution from data: concentration inequalities underpin the proof that the empirical CDF converges uniformly to the true CDF as the sample size grows. In the broader AI landscape, concentration inequalities are the theoretical foundation for generalisation bounds in statistical learning theory, which tell you how well a model trained on finite data will perform on unseen data.


## The mathematics

Concentration inequalities come from probability theory. This branch of mathematics deals with uncertainty: situations where an outcome is not fixed but has a range of possible values, each with a specific chance of occurring.

To understand these inequalities, we first need to agree on what we mean by "expectation." In everyday language, expectation might mean hope. In mathematics, it means the long-run average. If you repeat an experiment many times and take the average of the results, that average is the expectation.

For example, if you roll a fair six-sided die, the possible outcomes are 1, 2, 3, 4, 5, and 6. Each has a probability of 1/6. The expectation is calculated by multiplying each outcome by its probability and adding them up:

Expectation = (1 * 1/6) + (2 * 1/6) + (3 * 1/6) + (4 * 1/6) + (5 * 1/6) + (6 * 1/6)
            = 1/6 + 2/6 + 3/6 + 4/6 + 5/6 + 6/6
            = 21/6
            = 3.5

Even though you can never roll a 3.5, if you rolled the die thousands of times and averaged the results, the average would get closer and closer to 3.5.

A concentration inequality tells us how likely it is that a random quantity (like the average of several dice rolls) will stay close to its expectation. As we collect more data, the average tends to "concentrate" around the true expectation. These inequalities give us a mathematical guarantee on how tight that concentration is.

We will look at three such inequalities, moving from the simplest and loosest to the most powerful but restrictive.

1. Markov's Inequality
Markov's inequality is the starting point. It applies to any random quantity that is never negative (like a count of items, or a length). It says:

The probability that a non-negative random variable X is greater than some positive number 'a' is at most the expectation of X divided by 'a'.

In symbols: P(X ≥ a) ≤ E[X] / a.

This is a very loose bound. It requires almost no information about the distribution other than that it is non-negative and we know its average.

2. Chebyshev's Inequality
Chebyshev's inequality uses more information: the variance. Variance measures how spread out the values are around the mean. A high variance means the data points are far from the average; a low variance means they are clustered tightly.

Chebyshev's inequality says:

The probability that a random variable X deviates from its expectation E[X] by more than 'k' standard deviations is at most 1 divided by k squared.

In symbols: P(|X - E[X]| ≥ k * σ) ≤ 1 / k².

Here, σ (sigma) is the standard deviation, which is the square root of the variance. This bound is tighter than Markov's because it accounts for the spread of the data. However, it applies to any distribution with a defined variance, so it must be conservative.

3. Hoeffding's Inequality
Hoeffding's inequality is much stronger, but it has stricter requirements. It applies when we are averaging several independent random variables, and each variable is known to be bounded within a specific range (for example, between 0 and 1).

Suppose we have n independent samples, X₁, X₂, ..., Xₙ. Each sample falls between 0 and 1. Let the true average (expectation) of these samples be μ. Let the sample average be denoted by X-bar.

Hoeffding's inequality states:

The probability that the sample average X-bar differs from the true mean μ by more than 't' is at most 2 times e to the power of (-2 * n * t²).

In symbols: P(|X-bar - μ| ≥ t) ≤ 2 * exp(-2 * n * t²).

Here, 'exp' means the exponential function. The key feature is the term n in the exponent. As the sample size n grows, the value inside the exponent becomes a large negative number, making the entire probability bound shrink exponentially fast. This explains why more data leads to much more reliable estimates.

Note: A probability cannot be greater than 1. If the formula gives a number larger than 1, it simply means the bound is not useful (it tells us nothing we didn't already know, since probabilities are always ≤ 1). We only care when the bound is less than 1.


## A worked example

We will work through a concrete example to see how these bounds behave. Imagine we are drawing numbers from a uniform distribution between 0 and 1. This means every number between 0 and 1 is equally likely.

For a single draw from this distribution:
- The expectation (true mean) μ is 0.5.
- The variance is 1/12 ≈ 0.0833.
- The standard deviation σ is the square root of 1/12, which is approximately 0.2887.

We want to estimate the average of n draws. We will check the probability that this sample average differs from the true mean (0.5) by more than t = 0.1. In other words, we want to bound the probability that the average is outside the interval [0.4, 0.6].

Case 1: Small Sample Size (n = 10)

Step 1: Apply Markov's Inequality.
Markov's inequality applies to non-negative variables. Let S be the sum of the 10 draws. Since each draw is between 0 and 1, S is non-negative.
The expectation of the sum E[S] = n * μ = 10 * 0.5 = 5.
We want to bound the probability that the average is at least 0.6 above the mean? No, let's look at the upper tail first. If the average is ≥ 0.6, then the sum S ≥ 6.
Using Markov: P(S ≥ 6) ≤ E[S] / 6 = 5 / 6 ≈ 0.833.
This bound is very loose (83.3%). It tells us the probability is less than 83.3%, which is true but not helpful.

Step 2: Apply Chebyshev's Inequality.
First, we need the standard deviation of the *average* (X-bar), not the sum.
The variance of the average is Var(X-bar) = Var(Sum) / n² = (n * σ_individual²) / n² = σ_individual² / n.
Var(X-bar) = (1/12) / 10 = 1/120 ≈ 0.00833.
The standard deviation of the average, σ_avg, is sqrt(1/120) ≈ 0.0913.
We want the probability that |X-bar - 0.5| ≥ 0.1.
How many standard deviations is 0.1? k = t / σ_avg = 0.1 / 0.0913 ≈ 1.095.
Chebyshev bound: 1 / k² = 1 / (1.095)² ≈ 1 / 1.2 ≈ 0.833.
Again, the bound is ~83%. It is valid, but not tight.

Step 3: Apply Hoeffding's Inequality.
Hoeffding requires the variables to be bounded in [0, 1], which they are.
The formula is: P(|X-bar - μ| ≥ t) ≤ 2 * exp(-2 * n * t²).
Plug in n = 10 and t = 0.1:
Exponent = -2 * 10 * (0.1)² = -20 * 0.01 = -0.2.
Bound = 2 * exp(-0.2).
Using the value e^(-0.2) ≈ 0.819:
Bound = 2 * 0.819 = 1.638.
Since a probability cannot exceed 1, this bound is useless (it just says "the probability is less than 164%", which is always true). Hoeffding needs more data to become effective for this deviation.

Case 2: Medium Sample Size (n = 100)

Step 4: Apply Hoeffding's Inequality with n = 100.
Exponent = -2 * 100 * (0.1)² = -200 * 0.01 = -2.
Bound = 2 * exp(-2).
Using e^(-2) ≈ 0.135:
Bound = 2 * 0.135 = 0.27.
Now the bound is 27%. This means we are guaranteed that the probability of the average being off by more than 0.1 is less than 27%. This is a meaningful guarantee.

Case 3: Large Sample Size (n = 1000)

Step 5: Apply Hoeffding's Inequality with n = 1000.
Exponent = -2 * 1000 * (0.1)² = -2000 * 0.01 = -20.
Bound = 2 * exp(-20).
e^(-20) is an extremely small number, approximately 2.06 * 10^-9.
Bound ≈ 2 * 2.06 * 10^-9 ≈ 4.12 * 10^-9.
This is less than 0.000000005. It means the chance of the average being off by more than 0.1 is essentially zero.

Conclusion:
With only 10 samples, our bounds are loose or useless. With 100 samples, we have a reasonable guarantee (less than 27% error). With 1000 samples, the probability of significant error vanishes exponentially. This demonstrates the power of concentration inequalities: as n increases, the sample average becomes an increasingly precise estimate of the true mean.


## From scratch, in Python

```python
import random

# Step 1: Generate a sample of 1000 values from a uniform distribution between 0 and 1
sample = []
for i in range(1000):
    sample.append(random.random())

# Step 2: Compute the sample average (mean)
def mean(values):
    total = 0
    for v in values:
        total = total + v
    return total / len(values)

average = mean(sample)
print('Sample average:', average)

# Step 3: Compute the Hoeffding bound for this sample size and a deviation of 0.1
import math
t = 0.1
n = len(sample)
bound = 2 * math.exp(-2 * n * t * t)
print('Hoeffding bound for deviation', t, ':', bound)

# Step 4: Simulate many experiments to empirically check the bound
# Run 1000 times: draw 1000 uniform variables, compute average, check if it deviates from 0.5 by more than 0.1
count_large_deviation = 0
num_experiments = 1000
for experiment in range(num_experiments):
    sample = []
    for i in range(1000):
        sample.append(random.random())
    avg = mean(sample)
    if abs(avg - 0.5) > 0.1:
        count_large_deviation = count_large_deviation + 1

empirical_probability = count_large_deviation / num_experiments
print('Empirical probability of a deviation >0.1:', empirical_probability)
```


## What the code does

Step 1 creates an empty list called `sample`. A list is an ordered collection of items that can be added to. The `for` loop repeats the block inside 1000 times, each time generating a random number between 0 and 1 using `random.random()` and appending it to the list with `.append`. Step 2 defines a function called `mean` that takes a list, adds up all its values with a loop, and divides by how many there are. The trick is that `len(values)` gives the number of items in the list. Step 3 uses the `math.exp` function to compute the exponential, and the formula from earlier. Step 4 runs a simulation: it repeats the whole process 1000 times, each time checking if the average is more than 0.1 away from 0.5. The number of times this happens is divided by the total number of experiments to get an empirical frequency, which should be much lower than the Hoeffding bound. You will notice the bound is very loose — it says the probability is less than 4e-9, but in practice it might be even smaller. The point is that the bound is an upper limit, not an exact value.


## Common misconceptions

**Concentration inequalities tell you the exact probability** — If I use Hoeffding's inequality, I know exactly what the probability is of my average being off.

Concentration inequalities give upper bounds, not exact probabilities. The bound is often loose — the true probability could be much smaller. In fact: They are upper limits. They guarantee the probability is *at most* this value, not exactly this value.

**More data always helps, but the bound is not affected** — The sample size appears in the bound, but if I only double the data, the bound barely improves, so why bother?

The improvement is exponential in n for Hoeffding. Doubling n squares the exponent, which can be a dramatic improvement. In fact: For Hoeffding, the bound decays as exp(-2 n t^2), so increasing n rapidly reduces the probability of large deviations.

**All variables must be normally distributed** — I have to assume my data is normally distributed to use concentration inequalities.

Concentration inequalities are 'distribution-free' — they only require the variables to be independent and bounded (for Hoeffding), not a specific distribution. In fact: They work with any distribution, as long as the variables are independent and bounded between fixed values.

**If the bound is below 1, it is always useful** — As long as the right-hand side of the inequality is less than 1, it is a meaningful probability.

Sometimes the bound is, say, 0.9, which is nearly 1 — it says 'with probability at most 0.9', which is not informative because you already know any probability is ≤ 1. In fact: The bound is only useful when it is meaningfully below 1 (e.g., 0.05 or 0.01), and that requires a sufficiently large n or a small deviation relative to the spread.


## In short

Concentration inequalities are upper bounds on the probability that a random quantity (often a sample average) deviates from its expectation by more than a given amount. Markov's is the simplest and loosest; Chebyshev's uses variance for a better bound; Hoeffding's gives an exponential bound when variables are independent and bounded. They are the theoretical backbone of generalisation: they justify that a model trained on finite data performs well on unseen data, and they are used in every branch of applied ML to validate models, set thresholds, and decide when to trust an estimate. Remember: the bound is an upper limit, not an exact value; the bigger the sample size and the smaller the variability, the tighter the bound.


## Where this is used

Concentration inequalities are used everywhere in machine learning, though usually silently. Here is how they appear in practice, mechanism by mechanism.

**Large language models (LLMs).** When an LLM is trained on trillions of tokens, the training data is a huge sample from the distribution of human text. The loss function — how far the model's predictions are from the true next token — is an average over many examples. Concentration inequalities (specifically Bernstein or Hoeffding-style bounds) justify that the average loss on the training set is very close to the true expected loss, because the sample is so huge. This is why a model that performs well on its training data is expected to perform well on new, unseen text (up to the limit of overfitting). In practice, LLM trainers use these bounds to decide when the model has converged: during training, they monitor a small validation set, and if the average validation loss is within a few standard deviations of the training loss, they consider the model stable. The math is not written down in the training loop, but the principle underpins the entire enterprise.

**Small language models (on a phone or laptop).** When you run a small language model locally, it is often quantised to save memory. Quantisation adds noise to the weights. The model's outputs are then a random variable (due to the noise). Concentration inequalities are used to bound the effect of that noise: they tell the developer that with high probability, the output of the quantised model is close to the output of the original model. This is analysed in research on neural network quantisation, and it is why you can ship a model that is 4-bit quantised without its performance collapsing. The developer writes something like: 'the probability that the quantised model's loss differs from the float model's loss by more than 0.01 is less than 5%, given the noise bound from Hoeffding on the activations.'

**Retrieval-augmented generation (RAG).** RAG systems retrieve relevant documents from a database using embeddings (vectors that represent meaning). The retrieval step uses a similarity score (like cosine similarity) to rank documents. But the embeddings are learned and have uncertainty. Concentration inequalities are used to bound the probability that the retrieved set misses relevant documents: if the embedding of the query is a random vector (due to training noise), the probability that its similarity to the true relevant document is below a threshold is bounded by Chebyshev's inequality using the variance of the embeddings. In practice, developers set a threshold on retrieval scores, knowing that the false-negative rate is controlled by such a bound. When they fine-tune the embedding model, they monitor the validation retrieval accuracy, which is itself an average, and they use the concentration idea to decide if a change is statistically significant.

**Agentic systems that plan and call tools.** An AI agent might take a sequence of actions, each with a stochastic outcome (e.g., a tool call might fail or return noisy data). The agent has to decide whether to continue or stop. Concentration inequalities are used in the concept of 'probably approximately correct' (PAC) bounds on the agent's success rate: after N trials, the agent can estimate its success rate and bound the probability that the true success rate is far from the estimate. This is used in reinforcement learning, where an agent estimates the value of a state (the expected cumulative reward). The Q-learning update uses sample averages of rewards; concentration inequalities tell the agent how many times it must visit a state before it can trust its estimate to within a tolerance. This is the basis of the upper-confidence-bound (UCB) exploration strategy, where the agent balances exploration and exploitation: it chooses actions with high estimated value plus a confidence bonus that shrinks as more data is collected, and that bonus is exactly a concentration bound.

**Everyday applied machine learning (recommendation, vision, forecasting, fraud detection).** In any supervised learning task, you have a training set and a test set. A fundamental result in statistical learning theory says that the difference between the training error and the true error can be bounded by a concentration inequality (e.g., the Vapnik-Chervonenkis bound, which is a type of concentration). This is what allows you to say 'the model's accuracy on the test set is 93%, and I am 95% confident that its true accuracy is between 92% and 94%.' For a recommendation system, the average rating of an item is a sample average; the developer uses the standard error (which is derived from concentration) to decide how many ratings an item needs before it is displayed prominently. For fraud detection, the model's false-positive rate is estimated on a sample of transactions; concentration inequalities bound how close that estimate is to the true rate, which determines whether the fraud detection system meets regulatory requirements. In forecasting, prediction intervals are often constructed using concentration-type bounds, especially in conformal prediction, which gives distribution-free guarantees that the true value will be inside the interval with a stated probability.

What a developer actually writes: in machine learning code, you rarely call a function named `hoeffding_bound`, but you use the logic constantly. For example, in a validation loop, you compute the mean loss and a confidence interval. The interval is `mean ± z*std/sqrt(n)`, where `z` is a constant based on the normal approximation (which is itself a concentration result). You use this to decide whether a model is better than another model: you look at two confidence intervals; if they do not overlap, you conclude one is better. This is a direct application of concentration of the sample mean. If you did it naively without the bound, you might overreact to noise. So the developer's job is to compute the right bound and use it to calibrate their decisions. When it is done badly — e.g., ignoring the bound and trusting a sample of 10 ratings — the model makes poor decisions: a product with 10 five-star ratings looks like a hit, but that is just noise.


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Read this on the web: https://basavapurushottam.com/learn-ai/concentration-inequalities
